Added compiled presentation

latex/thesis/bibliography.bib

@@ -45,6 +45,15 @@
 %    eprint    = {https://doi.org/10.1080/24725854.2018.1550692}
 }
 
+@online{mackay_enc,
+    author = {MacKay, David J.C.},
+    title  = {Encyclopedia of Sparse Graph Codes},
+    date   = {2023-01},
+    url    = {http://www.inference.org.uk/mackay/codes/data.html}
+}
+
+
+
 @article{proximal_algorithms,
     author = {Parikh, Neal and Boyd, Stephen},
     title = {Proximal Algorithms},
@@ -210,16 +219,11 @@
   doi={10.1109/TIT.1962.1057683}
 }
 
-@online{lautern_channelcodes,
+@misc{lautern_channelcodes,
 	author = "Helmling, Michael and Scholl, Stefan and Gensheimer, Florian and Dietz, Tobias and Kraft, Kira and Ruzika, Stefan and Wehn, Norbert",
 	title = "{D}atabase of {C}hannel {C}odes and {ML} {S}imulation {R}esults",
+	howpublished = "\url{www.uni-kl.de/channel-codes}",
     url={https://www.uni-kl.de/channel-codes},
-	date = {2023-04}
+	year = "2023"
 }
 
-@online{mackay_enc,
-    author = {MacKay, David J.C.},
-    title  = {Encyclopedia of Sparse Graph Codes},
-    date   = {2023-04},
-    url    = {http://www.inference.org.uk/mackay/codes/data.html}
-}

latex/thesis/chapters/comparison.tex

@@ -2,9 +2,14 @@
 \label{chapter:comparison}
 
 In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
-First, the two algorithms are studied on a theoretical basis.
+First the two algorithms are compared on a theoretical basis.
-Subsequently, their respective simulation results are examined and their
+Subsequently, their respective simulation results are examined, and their
-differences are interpreted based on their theoretical structure.
+differences are interpreted on the basis of their theoretical structure.
+
+%some similarities between the proximal decoding algorithm
+%and \ac{LP} decoding using \ac{ADMM} are be pointed out.
+%The two algorithms are compared and their different computational and decoding
+%performance is interpreted on the basis of their theoretical structure.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -13,11 +18,12 @@ differences are interpreted based on their theoretical structure.
 
 \ac{ADMM} and the proximal gradient method can both be expressed in terms of
 proximal operators \cite[Sec. 4.4]{proximal_algorithms}.
-Additionally, the two algorithms show some striking similarities with
+When using \ac{ADMM} as an optimization method to solve the \ac{LP} decoding
-regard to their general structure and the way in which the minimization of the
+problem specifically, this is not quite possible because of the multiple
-respective objective functions is accomplished.
+constraints.
+In spite of that, the two algorithms still show some striking similarities.
 
-The \ac{LP} decoding problem in
+To see the first of these similarities, the \ac{LP} decoding problem in
 equation (\ref{eq:lp:relaxed_formulation}) can be slightly rewritten using the
 \textit{indicator functions} $g_j : \mathbb{R}^{d_j} \rightarrow
 \left\{ 0, +\infty \right\} \hspace{1mm}, j\in\mathcal{J}$ for the polytopes
@@ -34,11 +40,10 @@ by moving the constraints into the objective function, as shown in figure
 \ref{fig:ana:theo_comp_alg:admm}.
 Both algorithms are composed of an iterative approach consisting of two
 alternating steps.
-The objective functions of the two problems are similar in that they
+The objective functions of both problems are similar in that they
 both comprise two parts: one associated to the likelihood that a given
-codeword was sent, stemming from the channel model, and one associated
+codeword was sent and one associated to the constraints the decoding process
-to the constraints the decoding process is subjected to, stemming from the
+is subjected to.
-code used.
 %
 
 \begin{figure}[h]
@@ -118,11 +123,15 @@ Their major difference is that while with proximal decoding the constraints
 are regarded in a global context, considering all parity checks at the same
 time, with \ac{ADMM} each parity check is
 considered separately and in a more local context (line 4 in both algorithms).
+This difference means that while with proximal decoding the alternating
+minimization of the two parts of the objective function inevitably leads to
+oscillatory behavior (as explained in section
+\ref{subsec:prox:conv_properties}), this is not the case with \ac{ADMM}, which
+partly explains the disparate decoding performance of the two methods.
 Furthermore, while with proximal decoding the step considering the constraints
 is realized using gradient descent - amounting to an approximation -
 with \ac{ADMM} it reduces to a number of projections onto the parity polytopes
-$\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, which always provide exact
+$\mathcal{P}_{d_j}$ which always provide exact results.
-results.
 
 The contrasting treatment of the constraints (global and approximate with
 proximal decoding as opposed to local and exact with \ac{LP} decoding using
@@ -133,27 +142,34 @@ calculation, whereas with \ac{LP} decoding it occurs due to the approximate
 formulation of the constraints - independent of the optimization method
 itself. 
 The advantage which arises because of this when employing \ac{LP} decoding is
-the \ac{ML} certificate property: when a valid codeword is returned, it is
+that it can be easily detected \todo{Not 'easily' detected}, when the algorithm gets stuck - it
-also the \ac{ML} codeword.
+returns a solution corresponding to a pseudocodeword, the components of which
+are fractional.
+Moreover, when a valid codeword is returned, it is also the \ac{ML} codeword.
 This means that additional redundant parity-checks can be added successively
 until the codeword returned is valid and thus the \ac{ML} solution is found
 \cite[Sec. IV.]{alp}.
 
-In terms of time complexity the two decoding algorithms are comparable.
+In terms of time complexity, the two decoding algorithms are comparable.
 Each of the operations required for proximal decoding can be performed
-in $\mathcal{O}\left( n \right) $ time for \ac{LDPC} codes (see section
+in linear time for \ac{LDPC} codes (see section \ref{subsec:prox:comp_perf}).
-\ref{subsec:prox:comp_perf}).
+The same is true for the $\tilde{\boldsymbol{c}}$- and $\boldsymbol{u}$-update
-The same is true for \ac{LP} decoding using \ac{ADMM} (see section
+steps of \ac{LP} decoding using \ac{ADMM}, while
-\ref{subsec:admm:comp_perf}).
+the projection step has a worst-case time complexity of
-Additionally, both algorithms can be understood as message-passing algorithms,
+$\mathcal{O}\left( n^2 \right)$ and an average complexity of
-\ac{LP} decoding using \ac{ADMM} as similarly to
+$\mathcal{O}\left( n \right)$ (see section TODO, \cite[Sec. VIII.]{lautern}).
-\cite[Sec. III. D.]{original_admm} and
+
-\cite[Sec. II. B.]{efficient_lp_dec_admm}, and proximal decoding by starting
+Both algorithms can be understood as message-passing algorithms, \ac{LP}
-with algorithm \ref{alg:prox}, substituting for the gradient of the
+decoding using \ac{ADMM} as similarly to \cite[Sec. III. D.]{original_admm}
-code-constraint polynomial and separating the $\boldsymbol{s}$ update into two parts.
+or \cite[Sec. II. B.]{efficient_lp_dec_admm} and proximal decoding by
+starting with algorithm \ref{alg:prox},
+substituting for the gradient of the code-constraint polynomial and separating
+it into two parts.
 The algorithms in their message-passing form are depicted in figure
 \ref{fig:comp:message_passing}.
 $M_{j\to i}$ denotes a message transmitted from \ac{CN} j to \ac{VN} i.
+$M_{j\to}$ signifies the special case where a \ac{VN} transmits the same
+message to all \acp{VN}.
 %
 \begin{figure}[h]
     \centering
@@ -168,14 +184,14 @@ Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
 while stopping critierion unfulfilled do
     for j in $\mathcal{J}$ do
         $p_j \leftarrow \prod_{i\in N_c\left( j \right) } r_i $
-        $M_{j\to i} \leftarrow p_j^2 - p_j$|\Suppressnumber|
+        $M_{j\to} \leftarrow p_j^2 - p_j$|\Suppressnumber|
 |\vspace{0.22mm}\Reactivatenumber|
     end for
     for i in $\mathcal{I}$ do
         $s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
             \phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
                      |\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
-                        M_{j\to i} \right] $
+                        M_{j\to} \right] $
         $r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
     end for
 end while
@@ -221,9 +237,15 @@ return $\tilde{\boldsymbol{c}}$
     \label{fig:comp:message_passing}
 \end{figure}%
 %
-This message passing structure means that both algorithms can be implemented
+It is evident that while the two algorithms are very similar in their general
-very efficiently, as the update steps can be performed in parallel for all
+structure, with \ac{LP} decoding using \ac{ADMM}, multiple messages have to be
-\acp{CN} and for all \acp{VN}, respectively.
+computed for each check node (line 6 in figure
+\ref{fig:comp:message_passing:admm}), whereas
+with proximal decoding, the same message is transmitted to all \acp{VN}
+(line 5 of figure \ref{fig:comp:message_passing:proximal}).
+This means that while both algorithms have an average time complexity of
+$\mathcal{O}\left( n \right)$, more arithmetic operations are required in the
+\ac{ADMM} case.
 
 In conclusion, the two algorithms have a very similar structure, where the
 parts of the objective function relating to the likelihood and to the
@@ -231,8 +253,9 @@ constraints are minimized in an alternating fashion.
 With proximal decoding this minimization is performed for all constraints at once
 in an approximative manner, while with \ac{LP} decoding using \ac{ADMM} it is
 performed for each constraint individually and with exact results.
-In terms of time complexity, both algorithms are linear with
+In terms of time complexity, both algorithms are, on average, linear with
-respect to $n$ and are heavily parallelisable.
+respect to $n$, although for \ac{LP} decoding using \ac{ADMM} significantly
+more arithmetic operations are necessary in each iteration.
 
 
 
@@ -240,98 +263,24 @@ respect to $n$ and are heavily parallelisable.
 \section{Comparison of Simulation Results}%
 \label{sec:comp:res}
 
-The decoding performance of the two algorithms is compared in figure
+\begin{itemize}
-\ref{fig:comp:prox_admm_dec} in the form of the \ac{FER}.
+    \item The comparison of actual implementations is always debatable /
-Shown as well is the performance of the improved proximal decoding
+        contentious, since it is difficult to separate differences in
-algorithm presented in section \ref{sec:prox:Improved Implementation}.
+        algorithm performance from differences in implementation
-The \ac{FER} resulting from decoding using \ac{BP} and,
+    \item No large difference in computational performance $\rightarrow$
-wherever available, the \ac{FER} of \ac{ML} decoding taken from
+        Parallelism cannot come to fruition as decoding is performed on the
-\cite{lautern_channelcodes} are plotted as a reference.
+        same number of cores for both algorithms (Multiple decodings in parallel)
-The parameters chosen for the proximal and improved proximal decoders are
+    \item Nonetheless, in realtime applications / applications where the focus
-$\gamma=0.05$, $\omega=0.05$, $K=200$, $\eta = 1.5$ and $N=12$.
+        is not the mass decoding of raw data, \ac{ADMM} has advantages, since
-The parameters chosen for \ac{LP} decoding using \ac{ADMM} are $\mu = 5$,
+        the decoding of a single codeword is performed faster
-$\rho = 1$, $K=200$, $\epsilon_\text{pri} = 10^{-5}$ and
+    \item \ac{ADMM} faster than proximal decoding $\rightarrow$
-$\epsilon_\text{dual} = 10^{-5}$.
+        Parallelism
-For all codes considered in the scope of this work, \ac{LP} decoding using
+    \item Proximal decoding faster than \ac{ADMM} $\rightarrow$ dafuq
-\ac{ADMM} consistently outperforms both proximal decoding and the improved
+        (larger number of iterations before convergence? More values to compute for ADMM?)
-version, reaching very similar performance to \ac{BP}.
+\end{itemize}
-The decoding gain heavily depends on the code, evidently becoming greater for
-codes with larger $n$ and reaching values of up to $\SI{2}{dB}$.
 
-These simulation results can be interpreted with regard to the theoretical
-structure of the decoding methods, as analyzed in section \ref{sec:comp:theo}.
-The worse performance of proximal decoding is somewhat surprising, considering
-the global treatment of the constraints in contrast to the local treatment
-in the case of \ac{LP} decoding using \ac{ADMM}.
-It may be explained, however, in the context of the nature of the
-calculations performed in each case.
-With proximal decoding, the calculations are approximate, leading
-to the constraints never being quite satisfied.
-With \ac{LP} decoding using \ac{ADMM},
-the constraints are fulfilled for each parity check individualy after each
-iteration of the decoding process.
-It should be noted that while in this thesis proximal decoding was
-examined with respect to its performance in \ac{AWGN} channels, in
-\cite{proximal_paper} it is presented as a method applicable to non-trivial
-channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
-broadening its usefulness beyond what is shown here.
 
-The timing requirements of the decoding algorithms are visualized in figure
+\begin{figure}[H]
-\ref{fig:comp:time}.
-The datapoints have been generated by evaluating the metadata from \ac{FER}
-and \ac{BER} simulations using the parameters mentioned earlier when
-discussing the decoding performance.
-The codes considered are the same as in sections \ref{subsec:prox:comp_perf}
-and \ref{subsec:admm:comp_perf}.
-While the \ac{ADMM} implementation seems to be faster the the proximal
-decoding and improved proximal decoding implementations, infering some
-general behavior is difficult in this case.
-This is because of the comparison of actual implementations, making the
-results dependent on factors such as the grade of optimization of each of the
-implementations.
-Nevertheless, the run time of both the proximal decoding and the \ac{LP}
-decoding using \ac{ADMM} implementations is similar and both are
-reasonably performant, owing to the parallelisable structure of the
-algorithms.
-%
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[grid=both,
-                     xlabel={$n$}, ylabel={Time per frame (ms)},
-                     width=0.6\textwidth,
-                     height=0.45\textwidth,
-                     legend style={at={(0.5,-0.52)},anchor=south},
-                     legend cell align={left},]
-            \addplot[RedOrange, only marks, mark=square*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/proximal/fps_vs_n.csv};
-            \addlegendentry{Proximal decoding}
-
-            \addplot[Gray, only marks, mark=*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/hybrid/fps_vs_n.csv};
-            \addlegendentry{Improved proximal decoding ($N=12$)}
-
-            \addplot[NavyBlue, only marks, mark=triangle*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/admm/fps_vs_n.csv};
-            \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
-        \end{axis}    
-    \end{tikzpicture}
-
-    \caption{Comparison of the timing requirements of the different decoder implementations}
-    \label{fig:comp:time}
-\end{figure}%
-%
-\footnotetext{asdf}
-%
-
-\begin{figure}[h]
     \centering
     
     \begin{subfigure}[t]{0.48\textwidth}
@@ -350,19 +299,13 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_963965.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_963965.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         %{res/hybrid/2d_ber_fer_dfr_963965.csv};
                         {res/admm/ber_2d_963965.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle]
                     table [col sep=comma, x=SNR, y=FER,]
                         {res/generic/fer_ml_9633965.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_963965.csv};
             \end{axis}
         \end{tikzpicture}
 
@@ -386,20 +329,15 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_bch_31_26.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle*]
                     table [x=SNR, y=FER, col sep=comma,
                            discard if gt={SNR}{5.5},
-                           discard if lt={SNR}{1},]
+                           discard if lt={SNR}{1},
+                           ]
                         {res/generic/fer_ml_bch_31_26.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_bch_31_26.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -426,22 +364,15 @@ algorithms.
                            discard if not={gamma}{0.05},
                            discard if gt={SNR}{5.5}]
                         {res/proximal/2d_ber_fer_dfr_20433484.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05},
-                           discard if gt={SNR}{5.5}]
-                        {res/hybrid/2d_ber_fer_dfr_20433484.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma,
                            discard if not={mu}{3.0},
                            discard if gt={SNR}{5.5}]
                         {res/admm/ber_2d_20433484.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle, solid]
                     table [col sep=comma, x=SNR, y=FER,
                            discard if gt={SNR}{5.5}]
                         {res/generic/fer_ml_20433484.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_20433484.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -465,16 +396,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_20455187.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_20455187.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_20455187.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt,
-                          discard if gt={SNR}{5}]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_20455187.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -500,16 +424,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_40833844.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_40833844.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_40833844.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt,
-                          discard if gt={SNR}{3}]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_40833844.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -533,16 +450,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_pegreg252x504.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma,
-                           discard if gt={SNR}{3}]
-                        {res/generic/bp_pegreg252x504.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -560,21 +470,14 @@ algorithms.
                          xmin=10, xmax=50,
                          ymin=0, ymax=0.4,
                          legend columns=1,
-                         legend cell align={left},
                          legend style={draw=white!15!black}]
                 \addlegendimage{RedOrange, line width=1pt, mark=*, solid}
                 \addlegendentry{Proximal decoding}
-                
-                \addlegendimage{RedOrange, line width=1pt, mark=triangle, densely dashed}
-                \addlegendentry{Improved proximal decoding}
                
-                \addlegendimage{Turquoise, line width=1pt, mark=*}
+                \addlegendimage{NavyBlue, line width=1pt, mark=triangle, densely dashed}
                 \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
                 
-                \addlegendimage{RoyalPurple, line width=1pt, mark=*, solid}
+                \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
-                \addlegendentry{\acs{BP} (20 iterations)}
-                
-                \addlegendimage{Black, line width=1pt, mark=*, solid}
                 \addlegendentry{\acs{ML} decoding}
             \end{axis}
         \end{tikzpicture}
@@ -585,3 +488,32 @@ algorithms.
     \label{fig:comp:prox_admm_dec}
 \end{figure}
 
+\begin{figure}[h]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[grid=both,
+                     xlabel={$n$}, ylabel={Time per frame (s)},
+                     width=0.6\textwidth,
+                     height=0.45\textwidth,
+                     legend style={at={(0.5,-0.42)},anchor=south},
+                     legend cell align={left},]
+
+            \addplot[RedOrange, only marks, mark=*]
+                table [col sep=comma, x=n, y=spf]
+                    {res/proximal/fps_vs_n.csv};
+            \addlegendentry{Proximal decoding}
+            \addplot[PineGreen, only marks, mark=triangle*]
+                table [col sep=comma, x=n, y=spf]
+                    {res/admm/fps_vs_n.csv};
+            \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
+        \end{axis}
+    \end{tikzpicture}
+
+    \caption{Timing requirements of the proximal decoding imlementation%
+        \protect\footnotemark{}}
+    \label{fig:comp:time}
+\end{figure}%
+%
+\footnotetext{asdf}
+%

latex/thesis/chapters/conclusion.tex

@@ -1,55 +1,8 @@
 \chapter{Conclusion}%
 \label{chapter:conclusion}
 
-In the context of this thesis, two decoding algorithms were considered:
+\begin{itemize}
-proximal decoding and \ac{LP} decoding using \ac{ADMM}.
+    \item Summary of results
-The two algorithms were first analyzed individually, before comparing them
+    \item Future work
-based on simulation results as well as their theoretical structure.
+\end{itemize}
-
-For proximal decoding, the effect of each parameter on the behavior of the
-decoder was examined, leading to an approach to choosing the value of each
-of the parameters.
-The convergence properties of the algorithm were investigated in the context
-of the relatively high decoding failure rate, to derive an approach to correct
-possible wrong componets of the estimate.
-Based on this approach, an improvement over proximal decoding was suggested,
-leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
-the parameters considered.
-
-For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the
-relaxation while formulating the \ac{LP} decoding problem were first explored.
-The decomposable nature arising from the relocation of the constraints into
-the objective function itself was recognized as the major driver in enabling
-the efficent implementation of the decoding algorithm.
-Based on simulation results, general guidelines for choosing each parameter
-were again derived.
-The decoding performance, in form of the \ac{FER}, of the algorithm was
-analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that
-of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
-in question.
-
-Finally, strong parallells were discovered with regard to the theoretical
-structure of the two algorithms, both in the constitution of their respective
-objective functions as in the iterative approaches used to minimize them.
-One difference noted was the approximate nature of the minimization in the
-case of proximal decoding, leading to the constraints never being truly
-satisfied.
-In conjunction with the alternating minimization with respect to the same
-variable leading to oscillatory behavior, this was identified as the
-root cause of its comparatively worse decoding performance.
-Furthermore, both algorithms were expressed as message passing algorithms,
-justifying their similar computational performance.
-
-While the modified proximal decoding algorithm presented in section
-\ref{sec:prox:Improved Implementation} shows some promising results, further
-investigation is required to determine how different choices of parameters
-affect the decoding performance.
-Additionally, a more mathematically rigorous foundation for determining the
-potentially wrong components of the estimate is desirable.
-Another area benefiting from future work is the expantion of the \ac{ADMM}
-based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
-using \textit{adaptive \ac{LP} decoding}.
-With this method, the successive addition of redundant parity checks is used
-to mitigate the decoder becoming stuck in erroneous solutions introduced due
-the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}.
 

latex/thesis/chapters/discussion.tex

@@ -33,6 +33,13 @@ examined with respect to its performance in \ac{AWGN} channels, in
 channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
 broadening its usefulness beyond what is shown here.
 
+While the modified proximal decoding algorithm presented in section
+\ref{sec:prox:Improved Implementation} shows some promising results, further
+investigation is required to determine how different choices of parameters
+affect the decoding performance.
+Additionally, a more mathematically rigorous foundation for determining the
+potentially wrong components of the estimate is desirable.
+
 Another interesting approach might be the combination of proximal and \ac{LP}
 decoding.
 Performing an initial number of iterations using proximal decoding to obtain

latex/thesis/chapters/lp_dec_using_admm.tex

@@ -663,20 +663,20 @@ The same is true for the updates of the individual components of $\tilde{\boldsy
 This representation can be slightly simplified by substituting
 $\boldsymbol{\lambda}_j = \mu \cdot \boldsymbol{u}_j \,\forall\,j\in\mathcal{J}$:%
 %
-\begin{alignat}{3}
+\begin{alignat*}{3}
     \tilde{c}_i &\leftarrow \frac{1}{d_i} \left( 
         \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
             - \left( \boldsymbol{u}_j \right)_i \Big)
         - \frac{\gamma_i}{\mu} \right)
-    \hspace{3mm} && \forall i\in\mathcal{I} \label{eq:admm:c_update}\\
+    \hspace{3mm} && \forall i\in\mathcal{I} \\
     \boldsymbol{z}_j &\leftarrow \Pi_{\mathcal{P}_{d_j}}\left(
         \boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right)
-    \hspace{3mm} && \forall j\in\mathcal{J} \label{eq:admm:z_update}\\
+    \hspace{3mm} && \forall j\in\mathcal{J} \\
     \boldsymbol{u}_j &\leftarrow \boldsymbol{u}_j
         + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
         - \boldsymbol{z}_j
-    \hspace{3mm} && \forall j\in\mathcal{J} \label{eq:admm:u_update}
+    \hspace{3mm} && \forall j\in\mathcal{J}
-.\end{alignat}
+.\end{alignat*}
 %
 
 The reason \ac{ADMM} is able to perform so well is due to the relocation of the constraints
@@ -690,6 +690,7 @@ This can also be understood by interpreting the decoding process as a message-pa
 algorithm \cite[Sec. III. D.]{original_admm}, \cite[Sec. II. B.]{efficient_lp_dec_admm},
 depicted in algorithm \ref{alg:admm}.
 \todo{How are the variables being initialized?}
+\todo{Overrelaxation}
 
 \begin{genericAlgorithm}[caption={\ac{LP} decoding using \ac{ADMM} interpreted
             as a message passing algorithm\protect\footnotemark{}}, label={alg:admm},
@@ -726,21 +727,12 @@ a check-node update step (lines $3$-$6$) and the $\tilde{c}_i$-updates can be un
 a variable-node update step (lines $7$-$9$ in figure \ref{alg:admm}).
 The updates for each variable- and check-node can be perfomed in parallel.
 
-A technique called \textit{over-relaxation} can be employed to further improve
-convergence, introducing the over-relaxation parameter $\rho$.
-This consists of computing the term
-$\rho \boldsymbol{T}_j \tilde{\boldsymbol{c}} - \left( 1 - \rho \right)\boldsymbol{z}_j$
-before the $\boldsymbol{z}_j$ and $\boldsymbol{u}_j$ update steps (lines 4 and
-5 of algorithm \ref{alg:admm}) and
-subsequently replacing $\boldsymbol{T}_j \tilde{\boldsymbol{c}}$ with the
-computed value in the two updates \cite[Sec. 3.4.3]{distr_opt_book}.
-
 The main computational effort in solving the linear program then amounts to
 computing the projection operation $\Pi_{\mathcal{P}_{d_j}} \left( \cdot \right) $
 onto each check polytope. Various different methods to perform this projection
 have been proposed (e.g., in \cite{original_admm}, \cite{efficient_lp_dec_admm},
 \cite{lautern}).
-The method chosen here is the one presented in \cite{original_admm}.
+The method chosen here is the one presented in \cite{lautern}.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -801,7 +793,6 @@ Defining%
     \boldsymbol{s} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
         \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)
 \end{align*}%
-\todo{Rename $\boldsymbol{D}$}%
 %
 the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
 %
@@ -810,6 +801,7 @@ the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
         \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right)  
 .\end{align*}
 %
+
 This modified version of the decoding process is depicted in algorithm \ref{alg:admm:mod}.
 
 \begin{genericAlgorithm}[caption={\ac{LP} decoding using \ac{ADMM} algorithm with rewritten
@@ -843,41 +835,137 @@ return $\tilde{\boldsymbol{c}}$
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Analysis and Simulation Results}%
+\section{Results}%
-\label{sec:lp:Analysis and Simulation Results}
+\label{sec:lp:Results}
 
-In this section, \ac{LP} decoding using \ac{ADMM} is examined based on
-simulation results for various codes.
-First, the effect of the different parameters and how their values should be
-chosen is investigated.
-Subsequently, the decoding performance is observed and compared to that of
-\ac{BP}.
-Finally, the computational performance of the implementation and time
-complexity of the algorithm are studied.
 
-\subsection{Choice of Parameters}
+%\begin{figure}[H]
+%    \centering
+%
+%    \begin{tikzpicture}
+%        \begin{axis}[
+%            colormap/viridis,
+%            xlabel={$E_b / N_0$}, ylabel={$\mu$}, zlabel={\acs{BER}},
+%            view={75}{30},
+%            zmode=log,
+%        ]
+%            \addplot3[
+%                surf, 
+%                mesh/rows=14, mesh/cols=18
+%            ]
+%                table [col sep=comma, x=SNR, y=mu, z=BER]
+%                    {res/admm/ber_2d_20433484.csv};
+%        \end{axis}
+%    \end{tikzpicture}
+%\end{figure}
+%
+%\begin{figure}[H]
+%    \centering
+%
+%    \begin{tikzpicture}
+%        \begin{axis}[
+%            colormap/viridis,
+%            xlabel={$E_b / N_0 2$}, ylabel={$\mu$}, zlabel={\acs{BER}},
+%            view={75}{30},
+%            zmode=log,
+%        ]
+%            \addplot3[
+%                surf, 
+%                mesh/rows=14, mesh/cols=18
+%            ]
+%                table [col sep=comma, x=SNR, y=mu, z=BER]
+%                    {/home/andreas/git/ba_sw/scripts/admm/sim_results/ber_2d_20433484.csv};
+%        \end{axis}
+%    \end{tikzpicture}
+%\end{figure}
 
-The first two parameters to be investigated are the penalty parameter $\mu$
-and the over-relaxation parameter $\rho$. 
-A first approach to get some indication of the values that might be chosen
-for these parameters is to look at how the decoding performance depends
-on them.
-The \ac{FER} is plotted as a function of $\mu$ and $\rho$ in figure
-\ref{fig:admm:mu_rho}, for three different \acp{SNR}.
-The code chosen for this examination is a (3,6) regular \ac{LDPC} code with
-$n=204$ and $k=102$ \cite[\text{204.33.484}]{mackay_enc}.
-When varying $\mu$, $\rho$ is set to 1 and when varying
-$\rho$, $\mu$ is set to 5.
-$K$ is set to 200 and $\epsilon_\text{dual}$ and $\epsilon_\text{pri}$ to
-$10^{-5}$.
-The behavior that can be observed is very similar to that of the
-parameter $\gamma$ in proximal decoding, analyzed in section
-\ref{sec:prox:Analysis and Simulation Results}.
-A single optimal value giving optimal performance does not exist; rather,
-as long as the value is chosen within a certain range, the performance is
-approximately equally good.
 
-\begin{figure}[h]
+\textbf{Game Plan}
+
+\begin{enumerate}
+    \item Determine parameters
+    \item Make non-regular admm implementation use indexing instead of matrix vector
+        multiplication and simulate the pegreg, bch and 204.55.187 codes (parameter choice)
+    \item Computational performance
+    \item Comparison of proximal and admm (decoding performance and computational performance)
+    \item Find different codewords
+    \item Examine weird behavior when c is allowed to be negative
+    \item BP as comparison
+    \item Combination of proximal and BP
+\end{enumerate}
+
+
+\begin{itemize}
+    \item Choice of Parameters (Take decomposition paper as guide)
+        \begin{itemize}
+            \item mu
+            \item K
+            \item rho
+            \item epsilon pri / epslion dual
+        \end{itemize}
+    \item Decoding Performance
+        \begin{itemize}
+            \item FER and BER similar
+            \item DFR and FER pratically identical -> FER may be due to DFR
+            \item Compare to BP
+        \end{itemize}
+    \item Convergence Behavior
+        \begin{itemize}
+            \item Plot average error
+            \item Find out if converging to pseudocodeword or not converging.
+                How does pseudocodeword and not pseudocodeword relate to rounding and clipping?
+        \end{itemize}
+    \item Computational Performance
+        \begin{itemize}
+            \item Linear? Difficult to verify due to difference in adjustments
+                in the implementation for different codes
+        \end{itemize}
+
+\end{itemize}
+
+
+\begin{figure}[H]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[
+            grid=both,
+            xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={\acs{FER}},
+            ymode=log,
+            width=0.6\textwidth,
+            height=0.45\textwidth,
+            legend style={at={(0.5,-0.57)},anchor=south},
+        ]
+            \addplot[RedOrange, line width=1pt, mark=*]
+                table [col sep=comma, x=SNR, y=FER,
+                       discard if gt={SNR}{2.2},
+                       ]
+                    {res/admm/fer_paper_margulis.csv};
+            \addlegendentry{\acs{ADMM} (Barman et al.)}
+            \addplot[NavyBlue, densely dashed, line width=1pt, mark=triangle]
+                table [col sep=comma, x=SNR, y=FER,]
+                    {res/admm/ber_margulis264013203.csv};
+            \addlegendentry{\acs{ADMM} (Own results)}
+            \addplot[RoyalPurple, line width=1pt, mark=triangle]
+                table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},]
+                    {res/generic/fer_bp_mackay_margulis.csv};
+            \addlegendentry{\acs{BP} (Barman et al.)}
+        \end{axis}
+    \end{tikzpicture}
+    
+    \caption{Comparison of datapoints from Barman et al. with own simulation results%
+        \protect\footnotemark{}}
+    \label{fig:admm:results}
+\end{figure}%
+%
+\footnotetext{``Margulis'' \ac{LDPC} code with $n = 2640$, $k = 1320$
+    \cite[\text{Margulis2640.1320.3}]{mackay_enc}; $K=200, \mu = 3.3, \rho=1.9,
+        \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$
+}%
+%
+
+
+\begin{figure}[H]
     \centering
 
     \begin{subfigure}[c]{0.48\textwidth}
@@ -941,7 +1029,6 @@ approximately equally good.
             \begin{axis}[hide axis,
                          xmin=10, xmax=50,
                          ymin=0, ymax=0.4,
-                         legend columns=3,
                          legend style={draw=white!15!black,legend cell align=left}]
 
                 \addlegendimage{ForestGreen, line width=1pt, densely dashed, mark=*}
@@ -954,95 +1041,51 @@ approximately equally good.
         \end{tikzpicture}
     \end{subfigure}
 
-    \caption{Dependence of the decoding performance on the parameters $\mu$ and $\rho$.
+    \caption{asf}
-        (3,6) regular \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
-    \label{fig:admm:mu_rho}
-\end{figure}%
-
-To aid in the choice of the parameters, an additional criterion can be used:
-the number of iterations performed for a decoding operation.
-This is directly related to the time needed to decode a received vector
-$\boldsymbol{y}$, which the aim is to minimize.
-Figure \ref{fig:admm:mu_rho_iterations} shows the average number of iterations
-over $\SI{1000}{}$ decodings, as a function of $\rho$.
-This time the \ac{SNR} is kept constant at $\SI{4}{dB}$ and the parameter
-$\mu$ is varied.
-The values chosen for the rest of the parameters are the same as before.
-It is visible that choosing a large value for $\rho$ as well as a small value
-for $\mu$ minimizes the average number of iterations and thus the average
-run time of the decoding process.
-%
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[
-            grid=both,
-            xlabel={$\rho$}, ylabel={Average \# of iterations},
-            ymode=log,
-            width=0.6\textwidth,
-            height=0.45\textwidth,
-        ]
-            \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                table [col sep=comma, x=rho, y=k_avg,
-                       discard if not={mu}{9.0},]
-                    {res/admm/mu_rho_kavg_20433484.csv};
-            \addlegendentry{$\mu = 9$}
-            \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
-                table [col sep=comma, x=rho, y=k_avg,
-                       discard if not={mu}{5.0},]
-                    {res/admm/mu_rho_kavg_20433484.csv};
-            \addlegendentry{$\mu = 5$}
-            \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
-                table [col sep=comma, x=rho, y=k_avg,
-                       discard if not={mu}{2.0},]
-                    {res/admm/mu_rho_kavg_20433484.csv};
-            \addlegendentry{$\mu = 2$}
-        \end{axis}
-    \end{tikzpicture}
-
-    \caption{Dependence of the average number of iterations required on $\mu$ and $\rho$
-        for $E_b / N_0 = \SI{4}{dB}$. (3,6) regular \ac{LDPC} code with $n=204, k=102$
-        \cite[\text{204.33.484}]{mackay_enc}}
-    \label{fig:admm:mu_rho_iterations}
 \end{figure}%
 %
-The same behavior can be observed when looking at a number of different codes,
+%\footnotetext{(3,6) regular \ac{LDPC} code with $n = 204$, $k = 102$
-as shown in figure \ref{fig:admm:mu_rho_multiple}.
+%    \cite[\text{204.33.484}]{mackay_enc}; $K=200, \rho=1, \epsilon_\text{pri} = 10^{-5},
+%        \epsilon_\text{dual} = 10^{-5}$
+%}%
 %
-\begin{figure}[h]
+
+\begin{figure}[H]
     \centering
 
-    \begin{subfigure}[t]{0.48\textwidth}
+    \begin{subfigure}[c]{0.48\textwidth}
         \centering
+    
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$\rho$}, ylabel={Average \# of iterations},
+                xlabel={$\mu$}, ylabel={Average \# of iterations},
                 ymode=log,
                 width=\textwidth,
                 height=0.75\textwidth,
             ]
-                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_963965.csv};
-                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_963965.csv};
                 \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
+                    table [col sep=comma, x=mu, y=k_avg,
-                           discard if not={mu}{2.0},]
+                           discard if not={rho}{0.5000000000000001},]
-                        {res/admm/mu_rho_kavg_963965.csv};
+                        {res/admm/mu_rho_kavg_20433484.csv};
+                \addlegendentry{$\rho = 0.5$}
+                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
+                    table [col sep=comma, x=mu, y=k_avg,
+                           discard if not={rho}{1.1000000000000003},]
+                        {res/admm/mu_rho_kavg_20433484.csv};
+                \addlegendentry{$\rho = 1.1$}
+                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
+                    table [col sep=comma, x=mu, y=k_avg,
+                           discard if not={rho}{1.9000000000000004},]
+                        {res/admm/mu_rho_kavg_20433484.csv};
+                \addlegendentry{$\rho = 1.9$}
             \end{axis}
         \end{tikzpicture}
-        \caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
-            \cite[\text{96.3.965}]{mackay_enc}}
     \end{subfigure}%
-    \hfill
+    \hfill%
-    \begin{subfigure}[t]{0.48\textwidth}
+    \begin{subfigure}[c]{0.48\textwidth}
         \centering
+    
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
@@ -1051,181 +1094,30 @@ as shown in figure \ref{fig:admm:mu_rho_multiple}.
                 width=\textwidth,
                 height=0.75\textwidth,
             ]
-                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_bch_31_26.csv};
-                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_bch_31_26.csv};
-                \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{2.0},]
-                        {res/admm/mu_rho_kavg_bch_31_26.csv};
-            \end{axis}
-        \end{tikzpicture}
-        \caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
-    \end{subfigure}
-
-    \vspace{3mm}
-
-    \begin{subfigure}[t]{0.48\textwidth}
-        \centering
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\rho$}, ylabel={Average \# of iterations},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-            ]
-                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_20433484.csv};
-                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_20433484.csv};
                 \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
                     table [col sep=comma, x=rho, y=k_avg,
                            discard if not={mu}{2.0},]
                         {res/admm/mu_rho_kavg_20433484.csv};
-            \end{axis}
+                \addlegendentry{$\mu = 2$}
-        \end{tikzpicture}
-        \caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
-            \cite[\text{204.33.484}]{mackay_enc}}
-    \end{subfigure}%
-    \hfill
-    \begin{subfigure}[t]{0.48\textwidth}
-        \centering
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\rho$}, ylabel={Average \# of iterations},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-            ]
-                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_20455187.csv};
                 \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
                     table [col sep=comma, x=rho, y=k_avg,
                            discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_20455187.csv};
+                        {res/admm/mu_rho_kavg_20433484.csv};
-                \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
+                \addlegendentry{$\mu = 5$}
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{2.0},]
-                        {res/admm/mu_rho_kavg_20455187.csv};
-            \end{axis}
-        \end{tikzpicture}
-        \caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
-            \cite[\text{204.55.187}]{mackay_enc}}
-    \end{subfigure}%
-
-    \vspace{3mm}
-
-    \begin{subfigure}[t]{0.48\textwidth}
-        \centering
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\rho$}, ylabel={Average \# of iterations},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-            ]
                 \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
                     table [col sep=comma, x=rho, y=k_avg,
                            discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_40833844.csv};
+                        {res/admm/mu_rho_kavg_20433484.csv};
-                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
+                \addlegendentry{$\mu = 9$}
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_40833844.csv};
-                \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{2.0},]
-                        {res/admm/mu_rho_kavg_40833844.csv};
             \end{axis}
         \end{tikzpicture}
-        \caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
-            \cite[\text{408.33.844}]{mackay_enc}}
-    \end{subfigure}%
-    \hfill
-    \begin{subfigure}[t]{0.48\textwidth}
-        \centering
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\rho$}, ylabel={Average \# of iterations},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-            ]
-                \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{9.0},]
-                        {res/admm/mu_rho_kavg_pegreg252x504.csv};
-                \addplot[RedOrange, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{5.0},]
-                        {res/admm/mu_rho_kavg_pegreg252x504.csv};
-                \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
-                    table [col sep=comma, x=rho, y=k_avg,
-                           discard if not={mu}{2.0},]
-                        {res/admm/mu_rho_kavg_pegreg252x504.csv};
-            \end{axis}
-        \end{tikzpicture}
-        \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
-            \cite[\text{PEGReg252x504}]{mackay_enc}}
     \end{subfigure}%
 
-    \vspace{5mm}
+    \caption{asf}
+\end{figure}%
 
-    \begin{subfigure}[t]{\textwidth}
-        \centering
-        \begin{tikzpicture}
-            \begin{axis}[hide axis,
-                         xmin=10, xmax=50,
-                         ymin=0, ymax=0.4,
-                         legend style={draw=white!15!black,legend cell align=left}]
-                \addlegendimage{NavyBlue, line width=1.5pt, densely dashed, mark=*}
-                \addlegendentry{$\mu = 9$};
-                \addlegendimage{RedOrange, line width=1.5pt, densely dashed, mark=*}
-                \addlegendentry{$\mu = 5$};
-                \addlegendimage{ForestGreen, line width=1.5pt, densely dashed, mark=*}
-                \addlegendentry{$\mu = 2$};
-            \end{axis}
-        \end{tikzpicture}
 
-    \end{subfigure}
+\begin{figure}[H]
-
-    \caption{Dependence of the \ac{BER} on the value of the parameter $\gamma$ for various codes}
-    \label{fig:admm:mu_rho_multiple}
-\end{figure}
-
-To get an estimate for the parameter $K$, the average error during decoding
-can be used.
-This is shown in figure \ref{fig:admm:avg_error} as an average of
-$\SI{100000}{}$ decodings.
-$\mu$ is set to 5 and $\rho$ is set to $1$ and the rest of the parameters are
-again chosen as $K=200, \epsilon_\text{pri}=10^{-5}$ and $ \epsilon_\text{dual}=10^{-5}$.
-Similarly to the results in section
-\ref{sec:prox:Analysis and Simulation Results}, a dip is visible around the
-$20$ iteration mark.
-This is due to the fact that as the number of iterations increases
-more and more decodings converge, leaving only the mistaken ones to be
-averaged.
-The point at which the wrong decodings start to become dominant and the
-decoding performance does not increase any longer is largely independent of
-the \ac{SNR}, allowing the value of $K$ to be chosen without considering the
-\ac{SNR}.
-
-\begin{figure}[h]
     \centering
 
     \begin{tikzpicture}
@@ -1266,252 +1158,14 @@ the \ac{SNR}, allowing the value of $K$ to be chosen without considering the
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Average error for $\SI{100000}{}$ decodings. (3,6)
+    \caption{Average error for $\SI{100000}{}$ decodings\protect\footnotemark{}}
-        regular \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
+    \label{fig:}
-    \label{fig:admm:avg_error}
-\end{figure}%
-
-The last two parameters remaining to be examined are the tolerances for the
-stopping criterion of the algorithm, $\epsilon_\text{pri}$ and
-$\epsilon_\text{dual}$.
-These are both set to the same value $\epsilon$.
-The effect of their value on the decoding performance is visualized in figure
-\ref{fig:admm:epsilon}.
-All parameters except $\epsilon_\text{pri}$ and $\epsilon_\text{dual}$ are
-kept constant, with $K=200$, $\mu=5$, $\rho=1$ and $E_b / N_0 = \SI{4}{dB}$.
-A lower value for the tolerance initially leads to a dramatic decrease in the
-\ac{FER}, this effect fading as the tolerance becomes increasingly lower.
-
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[
-            grid=both,
-            xlabel={$\epsilon$}, ylabel={\acs{FER}},
-            ymode=log,
-            xmode=log,
-            x dir=reverse,
-            width=0.6\textwidth,
-            height=0.45\textwidth,
-        ]
-            \addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
-                table [col sep=comma, x=epsilon, y=FER,
-                       discard if not={SNR}{3.0},]
-                    {res/admm/fer_epsilon_20433484.csv};
-        \end{axis}
-    \end{tikzpicture}
-
-    \caption{Effect of the value of the parameters $\epsilon_\text{pri}$ and
-        $\epsilon_\text{dual}$ on the \acs{FER}. (3,6) regular \ac{LDPC} code with
-        $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
-    \label{fig:admm:epsilon}
-\end{figure}%
-
-In conclusion, the parameters $\mu$ and $\rho$ should be chosen comparatively
-small and large, respectively, to reduce the average runtime of the decoding
-process, while keeping them within a certain range as to not compromise the
-decoding performance.
-The maximum number of iterations $K$ performed can be chosen independantly
-of the \ac{SNR}.
-Finally, relatively small values should be given to the parameters
-$\epsilon_{\text{pri}}$ and $\epsilon_{\text{dual}}$ to achieve the lowest
-possible error rate.
-
-
-\subsection{Decoding Performance}
-
-In figure \ref{fig:admm:results}, the simulation results for the ``Margulis''
-\ac{LDPC} code ($n=2640$, $k=1320$) presented by Barman et al. in
-\cite{original_admm} are compared to the results from the simulations
-conducted in the context of this thesis.
-The parameters chosen were $\mu=3.3$, $\rho=1.9$, $K=1000$,
-$\epsilon_\text{pri}=10^{-5}$ and $\epsilon_\text{dual}=10^{-5}$,
-the same as in \cite{original_admm}.
-The two \ac{FER} curves are practically identical.
-Also shown is the curve resulting from \ac{BP} decoding, performing
-1000 iterations.
-The two algorithms perform relatively similarly, coming within $\SI{0.5}{dB}$
-of one another.
-
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[
-            grid=both,
-            xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={\acs{FER}},
-            ymode=log,
-            width=0.6\textwidth,
-            height=0.45\textwidth,
-            legend style={at={(0.5,-0.57)},anchor=south},
-            legend cell align={left},
-        ]
-            \addplot[Turquoise, line width=1pt, mark=*]
-                table [col sep=comma, x=SNR, y=FER,
-                       discard if gt={SNR}{2.2},
-                       ]
-                    {res/admm/fer_paper_margulis.csv};
-            \addlegendentry{\acs{ADMM} (Barman et al.)}
-            \addplot[NavyBlue, densely dashed, line width=1pt, mark=triangle]
-                table [col sep=comma, x=SNR, y=FER,]
-                    {res/admm/ber_margulis264013203.csv};
-            \addlegendentry{\acs{ADMM} (Own results)}
-            \addplot[RoyalPurple, line width=1pt, mark=*]
-                table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},]
-                    {res/generic/fer_bp_mackay_margulis.csv};
-            \addlegendentry{\acs{BP} (Barman et al.)}
-        \end{axis}
-    \end{tikzpicture}
-    
-    \caption{Comparison of datapoints from Barman et al. with own simulation results.
-        ``Margulis'' \ac{LDPC} code with $n = 2640$, $k = 1320$
-        \cite[\text{Margulis2640.1320.3}]{mackay_enc}\protect\footnotemark{}}
-    \label{fig:admm:results}
 \end{figure}%
 %
-In figure \ref{fig:admm:ber_fer}, the \ac{BER} and \ac{FER} for \ac{LP} decoding
+\footnotetext{(3,6) regular \ac{LDPC} code with $n = 204$, $k = 102$
-using\ac{ADMM} and \ac{BP} are shown for a (3, 6) regular \ac{LDPC} code with 
+    \cite[\text{204.33.484}]{mackay_enc}; $K=200, \rho=1, \epsilon_\text{pri} = 10^{-5},
-$n=204$.
+        \epsilon_\text{dual} = 10^{-5}$
-To ensure comparability, in both cases the number of iterations was set to
+}%
-$K=200$.
+%
-The values of the other parameters were chosen as $\mu = 5$, $\rho = 1$,
-$\epsilon = 10^{-5}$ and $\epsilon=10^{-5}$.
-Comparing figures \ref{fig:admm:results} and \ref{fig:admm:ber_fer} it is
-apparent that the difference in decoding performance depends on the code being
-considered.
-More simulation results are presented in figure \ref{fig:comp:prox_admm_dec}
-in section \ref{sec:comp:res}.
 
 
-\begin{figure}[h]
-    \centering
-
-    \begin{subfigure}[c]{0.48\textwidth}
-        \centering
-    
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\mu$}, ylabel={\acs{BER}},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-                ymax=1.5, ymin=3e-7,
-            ]
-                \addplot[Turquoise, line width=1pt, mark=*]
-                    table [col sep=comma, x=SNR, y=BER,
-                           discard if not={mu}{5.0},
-                           discard if gt={SNR}{4.5}]
-                        {res/admm/ber_2d_20433484.csv};
-                \addplot[RoyalPurple, line width=1pt, mark=*]
-                    table [col sep=comma, x=SNR, y=BER,
-                           discard if gt={SNR}{4.5}]
-                        {/home/andreas/bp_20433484.csv};
-            \end{axis}
-        \end{tikzpicture}
-    \end{subfigure}%
-    \hfill%
-    \begin{subfigure}[c]{0.48\textwidth}
-        \centering
-    
-        \begin{tikzpicture}
-            \begin{axis}[
-                grid=both,
-                xlabel={$\rho$}, ylabel={\acs{FER}},
-                ymode=log,
-                width=\textwidth,
-                height=0.75\textwidth,
-                ymax=1.5, ymin=3e-7,
-            ]
-                \addplot[Turquoise, line width=1pt, mark=*]
-                    table [col sep=comma, x=SNR, y=FER,
-                           discard if not={mu}{5.0},
-                           discard if gt={SNR}{4.5}]
-                        {res/admm/ber_2d_20433484.csv};
-                \addplot[RoyalPurple, line width=1pt, mark=*]
-                    table [col sep=comma, x=SNR, y=FER,
-                           discard if gt={SNR}{4.5}]
-                        {/home/andreas/bp_20433484.csv};
-            \end{axis}
-        \end{tikzpicture}
-    \end{subfigure}%
-
-    \begin{subfigure}[t]{\textwidth}
-        \centering
-
-        \begin{tikzpicture}
-            \begin{axis}[hide axis,
-                         xmin=10, xmax=50,
-                         ymin=0, ymax=0.4,
-                         legend columns=3,
-                         legend style={draw=white!15!black,legend cell align=left}]
-
-                \addlegendimage{Turquoise, line width=1pt, mark=*}
-                \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
-                \addlegendimage{RoyalPurple, line width=1pt, mark=*}
-                \addlegendentry{BP (200 iterations)}
-            \end{axis}
-        \end{tikzpicture}
-    \end{subfigure}
-
-    \caption{Comparison of the decoding performance of \acs{LP} decoding using
-        \acs{ADMM} and \acs{BP}. (3,6) regular \ac{LDPC} code with $n = 204$, $k = 102$
-        \cite[\text{204.33.484}]{mackay_enc}}
-    \label{fig:admm:ber_fer}
-\end{figure}%
-
-In summary, the decoding performance of \ac{LP} decoding using \ac{ADMM} comes
-close to that of \ac{BP}, their difference staying in the range of
-approximately $\SI{0.5}{dB}$, depending on the code in question.
-
-\subsection{Computational Performance}
-\label{subsec:admm:comp_perf}
-
-In terms of time complexity, the three steps of the decoding algorithm
-in equations (\ref{eq:admm:c_update}) - (\ref{eq:admm:u_update}) have to be
-considered.
-The $\tilde{\boldsymbol{c}}$- and $\boldsymbol{u}_j$-update steps are
-$\mathcal{O}\left( n \right)$ \cite[Sec. III. C.]{original_admm}.
-The complexity of the $\boldsymbol{z}_j$-update step depends on the projection
-algorithm employed.
-Since for the implementation completed for this work the projection algorithm
-presented in \cite{original_admm} is used, the $\boldsymbol{z}_j$-update step
-also has linear time complexity.
-
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[grid=both,
-                     xlabel={$n$}, ylabel={Time per frame (s)},
-                     width=0.6\textwidth,
-                     height=0.45\textwidth,
-                     legend style={at={(0.5,-0.42)},anchor=south},
-                     legend cell align={left},]
-
-            \addplot[NavyBlue, only marks, mark=triangle*]
-                table [col sep=comma, x=n, y=spf]
-                    {res/admm/fps_vs_n.csv};
-        \end{axis}
-    \end{tikzpicture}
-
-    \caption{Timing requirements of the \ac{LP} decoding using \ac{ADMM} implementation}
-    \label{fig:admm:time}
-\end{figure}%
-
-Simulation results from a range of different codes can be used to verify this
-analysis.
-Figure \ref{fig:admm:time} shows the average time needed to decode one
-frame as a function of its length.
-The codes used for this consideration are the same as in section \ref{subsec:prox:comp_perf}
-The results are necessarily skewed because these vary not only
-in their length, but also in their construction scheme and rate.
-Additionally, different optimization opportunities arise depending on the
-length of a code, since for smaller codes dynamic memory allocation can be
-completely omitted.
-This may explain why the datapoint at $n=504$ is higher then would be expected
-with linear behavior.
-Nonetheless, the simulation results roughly match the expected behavior
-following from the theoretical considerations.
-

latex/thesis/chapters/proximal_decoding.tex

@@ -300,7 +300,7 @@ the gradient can be written as%
 \begin{align*}
     \nabla h\left( \tilde{\boldsymbol{x}} \right) =
         4\left( \tilde{\boldsymbol{x}}^{\circ 3} - \tilde{\boldsymbol{x}} \right)
-        + 2\tilde{\boldsymbol{x}}^{\circ \left( -1 \right) } \circ \boldsymbol{H}^\text{T}
+        + 2\tilde{\boldsymbol{x}}^{\circ -1} \circ \boldsymbol{H}^\text{T}
             \boldsymbol{v}
 ,\end{align*}
 %
@@ -350,7 +350,7 @@ $\gamma$ are shown, as well as the curve resulting from decoding
 using a \ac{BP} decoder, as a reference.
 The results from Wadayama et al. are shown with solid lines,
 while the newly generated ones are shown with dashed lines.
-%
+
 \begin{figure}[h]
     \centering
 
@@ -359,7 +359,6 @@ while the newly generated ones are shown with dashed lines.
                      xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
                      ymode=log,
                      legend style={at={(0.5,-0.7)},anchor=south},
-                     legend cell align={left},
                      width=0.6\textwidth,
                      height=0.45\textwidth,
                      ymax=1.2, ymin=0.8e-4,
@@ -398,19 +397,21 @@ while the newly generated ones are shown with dashed lines.
             \addlegendentry{$\gamma = 0.05$ (Own results)}
 
             \addplot [RoyalPurple, mark=*, line width=1pt]
-                table [x=SNR, y=BER, col sep=comma,
+                table [x=SNR, y=BP, col sep=comma] {res/proximal/ber_paper.csv};
-                       discard if gt={SNR}{3.5}]
+            \addlegendentry{BP (Wadayama et al.)}
-                {res/generic/bp_20433484.csv};
-            \addlegendentry{BP (200 iterations)}
         \end{axis}
     \end{tikzpicture}
     
-    \caption{Comparison of datapoints from Wadayama et al. with own simulation results.
+    \caption{Comparison of datapoints from Wadayama et al. with own simulation results%
-        (3, 6) regular \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
+        \protect\footnotemark{}}
     \label{fig:prox:results}
 \end{figure}
 %
-It is noticeable that for a moderately chosen value of $\gamma$
+\footnotetext{(3,6) regular \ac{LDPC} code with $n = 204$, $k = 102$
+    \cite[\text{204.33.484}]{mackay_enc}; $\omega = 0.05, K=200, \eta=1.5$
+}%
+%
+\noindent It is noticeable that for a moderately chosen value of $\gamma$
 ($\gamma = 0.05$) the decoding performance is better than for low
 ($\gamma = 0.01$) or high ($\gamma = 0.15$) values.
 The question arises whether there is some optimal value maximizing the decoding
@@ -468,11 +469,14 @@ rather a certain interval in which it stays largely unchanged.
     \end{tikzpicture}
 
     \caption{Visualization of the relationship between the decoding performance
-        and the parameter $\gamma$. (3,6) regular \ac{LDPC} code with
+        \protect\footnotemark{} and the parameter $\gamma$}
-        $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
     \label{fig:prox:results_3d}
 \end{figure}%
 %
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $\omega = 0.05, K=200, \eta=1.5$
+}%
+%
 This indicates that while the choice of the parameter $\gamma$
 significantly affects the decoding performance, there is not much benefit
 attainable in undertaking an extensive search for an exact optimum.
@@ -534,11 +538,15 @@ depicted in figure \ref{fig:prox:gamma_omega_multiple}.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{The \ac{BER} as a function of the two step sizes. (3,6) regular
+    \caption{The \ac{BER} as a function of the two step sizes\protect\footnotemark{}}
-        \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
     \label{fig:prox:gamma_omega}
 \end{figure}%
 %
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $E_b / N_0=\SI{4}{dB}, K=100, \eta=1.5$
+}%
+%
+
 
 To better understand how to determine the optimal value for the parameter $K$,
 the average error is inspected.
@@ -594,10 +602,13 @@ stabilizes.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Average error for $\SI{500000}{}$ decodings. (3,6) regular \ac{LDPC} code
+    \caption{Average error for $\SI{500000}{}$ decodings\protect\footnotemark{}}
-        with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
 \end{figure}%
 %
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $\gamma=0.05, \omega = 0.05, K=200, \eta=1.5$
+}%
+%
 
 Changing the parameter $\eta$ does not appear to have a significant effect on
 the decoding performance when keeping the value within a reasonable window
@@ -691,11 +702,14 @@ means to bring about numerical stability.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Comparison of \ac{FER}, \ac{BER} and decoding failure rate. (3,6) regular
+    \caption{Comparison of \ac{FER}, \ac{BER} and decoding failure rate\protect\footnotemark{}}
-        \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
     \label{fig:prox:ber_fer_dfr}
 \end{figure}%
 %
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $\omega = 0.05, K=100, \eta=1.5$
+}%
+%
 
 Until now, only the \ac{BER} has been considered to gauge the decoding
 performance.
@@ -741,7 +755,7 @@ iteration ($\boldsymbol{r}$ and $\boldsymbol{s}$ are counted as different
 estimates and their values are interwoven to obtain the shown result), as well
 as the gradients of the negative log-likelihood and the code-constraint
 polynomial, which influence the next estimate.
-%
+
 \begin{figure}[h]
     \begin{subfigure}[t]{0.48\textwidth}
         \centering
@@ -947,11 +961,16 @@ polynomial, which influence the next estimate.
         \end{tikzpicture}
     \end{subfigure}
 
-    \caption{Visualization of a single decoding operation. BCH$\left( 7, 4 \right)$ code}
+    \caption{Visualization of a single decoding operation\protect\footnotemark{}
+        for a code with $n=7$}
     \label{fig:prox:convergence}
 \end{figure}%
+%
+\footnotetext{BCH$\left( 7,4 \right) $ code; $\gamma = 0.05, \omega = 0.05, K=200,
+    \eta = 1.5, E_b / N_0 = \SI{5}{dB}$
+}%
 % 
-It is evident that in all cases, past a certain number of
+\noindent It is evident that in all cases, past a certain number of
 iterations, the estimate starts to oscillate around a particular value.
 Jointly, the two gradients stop further approaching the value
 zero.
@@ -1137,11 +1156,16 @@ an invalid codeword.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Visualization of a single decoding operation. (3,6) regular \ac{LDPC} code
+    \caption{Visualization of a single decoding operation\protect\footnotemark{}
-        with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
+        for a code with $n=204$}
     \label{fig:prox:convergence_large_n}
 \end{figure}%
 %
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $\gamma=0.05, \omega = 0.05, K=200, \eta=1.5,
+    E_b / N_0 = \SI{5}{dB}$
+}%
+%
 
 
 \subsection{Computational Performance}
@@ -1167,10 +1191,6 @@ practical since it is the same as that of $\ac{BP}$.
 
 This theoretical analysis is also corroborated by the practical results shown
 in figure \ref{fig:prox:time_comp}.
-The codes considered are the BCH(31, 11) and BCH(31, 26) codes, a number of (3, 6)
-regular \ac{LDPC} codes (\cite[\text{96.3.965, 204.33.484, 408.33.844}]{mackay_enc}),
-a (5,10) regular \ac{LDPC} code (\cite[\text{204.55.187}]{mackay_enc}) and a
-progressive edge growth construction code (\cite[\text{PEGReg252x504}]{mackay_enc}).
 Some deviations from linear behavior are unavoidable because not all codes
 considered are actually \ac{LDPC} codes, or \ac{LDPC} codes constructed
 according to the same scheme.
@@ -1189,16 +1209,23 @@ $\SI{2.80}{GHz}$ and utilizing all cores.
                      height=0.45\textwidth,
                      legend cell align={left},]
 
-            \addplot[RedOrange, only marks, mark=square*]
+            \addplot[RedOrange, only marks, mark=*]
                 table [col sep=comma, x=n, y=spf]
                     {res/proximal/fps_vs_n.csv};
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Timing requirements of the proximal decoding imlementation}
+    \caption{Timing requirements of the proximal decoding imlementation%
+        \protect\footnotemark{}}
     \label{fig:prox:time_comp}
 \end{figure}%
 %
+\footnotetext{The datapoints depicted were calculated by evaluating the
+    metadata of \ac{FER} simulation results from the following codes:
+    BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187;
+    408.33.844; PEGReg252x504}]{mackay_enc}
+}%
+%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1230,7 +1257,7 @@ section \ref{subsec:prox:conv_properties} and shown in figure
 \ref{fig:prox:convergence_large_n}) and the probability of having a bit
 error are strongly correlated, a relationship depicted in figure
 \ref{fig:prox:correlation}.
-%
+  
 \begin{figure}[h]
     \centering
 
@@ -1253,17 +1280,22 @@ error are strongly correlated, a relationship depicted in figure
     \end{tikzpicture}
 
     \caption{Correlation between the occurrence of a bit error and the
-        amplitude of oscillation of the gradient of the code-constraint polynomial.
+        amplitude of oscillation of the gradient of the code-constraint polynomial%
-        (3,6) regular \ac{LDPC} code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
+        \protect\footnotemark{}}
     \label{fig:prox:correlation}
 \end{figure}%
 %
-The y-axis depicts whether there is a bit error and the x-axis the
+\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
+    \cite[\text{204.33.484}]{mackay_enc}; $\gamma = 0.05, \omega = 0.05, K=100, \eta=1.5$
+}%
+%
+
+\noindent The y-axis depicts whether there is a bit error and the x-axis the
 variance in $\nabla h\left( \tilde{\boldsymbol{x}} \right)$ after the
 100th iteration.
 While this is not exactly the magnitude of the oscillation, it is
 proportional and easier to compute.
-The datapoints are taken from a single decoding operation.
+The datapoints are taken from a single decoding operation
 
 Using this observation as a rule to determine the $N\in\mathbb{N}$ most
 probably wrong bits, all variations of the estimate with those bits modified
@@ -1447,9 +1479,7 @@ In some cases, a gain of up to $\SI{1}{dB}$ or higher can be achieved.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Comparison of the decoding performance between proximal decoding and the
+    \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
-        improved implementation. (3,6) regular \ac{LDPC} code with $n=204, k=102$
-        \cite[\text{204.33.484}]{mackay_enc}}
     \label{fig:prox:improved_results}
 \end{figure}
 
@@ -1479,12 +1509,12 @@ theoretical considerations.
                      legend style={at={(0.05,0.77)},anchor=south west},
                      legend cell align={left},]
 
-            \addplot[RedOrange, only marks, mark=square*]
+            \addplot[RedOrange, only marks, mark=*]
                 table [col sep=comma, x=n, y=spf]
                     {res/proximal/fps_vs_n.csv};
             \addlegendentry{proximal}
 
-            \addplot[Gray, only marks, mark=*]
+            \addplot[RoyalPurple, only marks, mark=triangle*]
                 table [col sep=comma, x=n, y=spf]
                     {res/hybrid/fps_vs_n.csv};
             \addlegendentry{improved ($N = 12$)}
@@ -1492,10 +1522,16 @@ theoretical considerations.
     \end{tikzpicture}
 
     \caption{Comparison of the timing requirements of the implementations of proximal
-        decoding and the improved algorithm}
+        decoding and the improved algorithm\protect\footnotemark{}}
     \label{fig:prox:time_complexity_comp}
 \end{figure}%
 %
+\footnotetext{The datapoints depicted were calculated by evaluating the
+    metadata of \ac{FER} simulation results from the following codes:
+    BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187;
+    408.33.844; PEGReg252x504}]{mackay_enc}
+}%
+%
 
 In conclusion, the decoding performance of proximal decoding can be improved
 by appending an ML-in-the-List step when the algorithm does not produce a

latex/thesis/res/admm/dfr_20433484.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,DFR,num_iterations
-1.0,0.06409994553376906,0.7013888888888888,0.7013888888888888,144.0
-1.5,0.03594771241830065,0.45495495495495497,0.47297297297297297,222.0
-2.0,0.014664163537755528,0.2148936170212766,0.2297872340425532,470.0
-2.5,0.004731522238525039,0.07634164777021919,0.08163265306122448,1323.0
-3.0,0.000911436423803915,0.016994783779236074,0.01749957933703517,5943.0
-3.5,0.00011736135227863285,0.002537369677176234,0.002587614621278734,39805.0
-4.0,1.0686274509803922e-05,0.00024,0.00023,100000.0
-4.5,4.411764705882353e-07,1e-05,1e-05,100000.0

latex/thesis/res/admm/dfr_20433484_metadata.json

@@ -1,10 +0,0 @@
-{
-    "duration": 46.20855645602569,
-    "name": "ber_20433484",
-    "platform": "Linux-6.2.10-arch1-1-x86_64-with-glibc2.37",
-    "K": 200,
-    "epsilon_pri": 1e-05,
-    "epsilon_dual": 1e-05,
-    "max_frame_errors": 100,
-    "end_time": "2023-04-22 19:15:37.252176"
-}

latex/thesis/res/admm/fer_epsilon_20433484.csv

@@ -1,31 +0,0 @@
-SNR,epsilon,BER,FER,DFR,num_iterations,k_avg
-1.0,1e-06,0.294328431372549,0.722,0.0,1000.0,0.278
-1.0,1e-05,0.294328431372549,0.722,0.0,1000.0,0.278
-1.0,0.0001,0.3059313725490196,0.745,0.0,1000.0,0.254
-1.0,0.001,0.3059558823529412,0.748,0.0,1000.0,0.254
-1.0,0.01,0.30637254901960786,0.786,0.0,1000.0,0.254
-1.0,0.1,0.31590686274509805,0.9,0.0,1000.0,0.925
-2.0,1e-06,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,1e-05,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,0.0001,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,0.001,0.11652450980392157,0.306,0.0,1000.0,0.702
-2.0,0.01,0.09901470588235294,0.297,0.0,1000.0,0.75
-2.0,0.1,0.10512745098039215,0.543,0.0,1000.0,0.954
-3.0,1e-06,0.004563725490196078,0.012,0.0,1000.0,0.988
-3.0,1e-05,0.004563725490196078,0.012,0.0,1000.0,0.988
-3.0,0.0001,0.005730392156862745,0.015,0.0,1000.0,0.985
-3.0,0.001,0.005730392156862745,0.015,0.0,1000.0,0.985
-3.0,0.01,0.007200980392156863,0.037,0.0,1000.0,0.981
-3.0,0.1,0.009151960784313726,0.208,0.0,1000.0,0.995
-4.0,1e-06,0.0,0.0,0.0,1000.0,1.0
-4.0,1e-05,0.0,0.0,0.0,1000.0,1.0
-4.0,0.0001,0.0,0.0,0.0,1000.0,1.0
-4.0,0.001,0.0002598039215686275,0.001,0.0,1000.0,0.999
-4.0,0.01,4.901960784313725e-05,0.01,0.0,1000.0,1.0
-4.0,0.1,0.0006862745098039216,0.103,0.0,1000.0,1.0
-5.0,1e-06,0.0,0.0,0.0,1000.0,1.0
-5.0,1e-05,0.0,0.0,0.0,1000.0,1.0
-5.0,0.0001,0.0,0.0,0.0,1000.0,1.0
-5.0,0.001,0.0,0.0,0.0,1000.0,1.0
-5.0,0.01,9.803921568627451e-06,0.002,0.0,1000.0,1.0
-5.0,0.1,0.0005245098039215687,0.097,0.0,1000.0,1.0

latex/thesis/res/admm/fer_epsilon_20433484_metadata.json

@@ -1,10 +0,0 @@
-{
-    "duration": 15.407989402010571,
-    "name": "rho_kavg_20433484",
-    "platform": "Linux-6.2.10-arch1-1-x86_64-with-glibc2.37",
-    "K": 200,
-    "rho": 1,
-    "mu": 5,
-    "max_frame_errors": 100000,
-    "end_time": "2023-04-23 06:39:23.561294"
-}

latex/thesis/res/generic/bp_20433484.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0315398614535635,0.598802395209581,334
-1.5,0.0172476397966594,0.352733686067019,567
-2,0.00668670591018522,0.14194464158978,1409
-2.5,0.00168951075575861,0.0388349514563107,5150
-3,0.000328745799468201,0.00800288103717338,24991
-3.5,4.21796065456326e-05,0.00109195935727272,183157
-4,4.95098039215686e-06,0.000134,500000
-4.5,2.94117647058824e-07,1.2e-05,500000

latex/thesis/res/generic/bp_20455187.csv

@@ -1,10 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0592497868712702,0.966183574879227,207
-1.5,0.0465686274509804,0.854700854700855,234
-2,0.0326898561282098,0.619195046439629,323
-2.5,0.0171613765211425,0.368324125230203,543
-3,0.00553787541776455,0.116346713205352,1719
-3.5,0.00134027952469441,0.0275900124155056,7249
-4,0.000166480738027721,0.0034201480924124,58477
-4.5,1.19607843137255e-05,0.000252,500000
-5,9.50980392156863e-07,2e-05,500000

latex/thesis/res/generic/bp_40833844.csv

@@ -1,6 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0303507766743061,0.649350649350649,308
-1.5,0.0122803195352215,0.296296296296296,675
-2,0.00284899516991547,0.0733137829912024,2728
-2.5,0.000348582879119279,0.00916968502131952,21811
-3,2.52493009763634e-05,0.000760274154860243,263063

latex/thesis/res/generic/bp_963965.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0352267331433998,0.56980056980057,351
-1.5,0.0214331413947537,0.383877159309021,521
-2,0.0130737396538751,0.225733634311512,886
-2.5,0.00488312520012808,0.0960614793467819,2082
-3,0.00203045475576707,0.0396589331746976,5043
-3.5,0.000513233833401836,0.010275380189067,19464
-4,0.000107190497363908,0.0025408117893667,78715
-4.5,3.2074433522095e-05,0.00092605883251763,215969

latex/thesis/res/generic/bp_bch_31_26.csv

@@ -1,11 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0584002878042931,0.743494423791822,269
-1.5,0.0458084740620892,0.626959247648903,319
-2,0.0318872821653689,0.459770114942529,435
-2.5,0.0248431459534825,0.392927308447937,509
-3,0.0158453558113146,0.251256281407035,796
-3.5,0.0115615186982586,0.176991150442478,1130
-4,0.00708558642550811,0.115606936416185,1730
-4.5,0.00389714705036542,0.0614817091915155,3253
-5,0.00221548053104734,0.0345125107851596,5795
-5.5,0.00106888328072207,0.0172131852999398,11619

latex/thesis/res/generic/bp_pegreg252x504.csv

@@ -1,6 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0294665331073098,0.647249190938511,309
-1.5,0.0104905437352246,0.265957446808511,752
-2,0.0021089358705197,0.05616399887672,3561
-2.5,0.000172515567971134,0.00498840196543037,40093
-3,6.3531746031746e-06,0.0002,500000

latex/thesis/thesis.tex

@@ -9,9 +9,9 @@
 \thesisTitle{Application of Optimization Algorithms for Channel Decoding}
 \thesisType{Bachelor's Thesis}
 \thesisAuthor{Andreas Tsouchlos}
-\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
+%\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
 %\thesisHeadOfInstitute{Prof. Dr.-Ing. Laurent Schmalen}
-\thesisSupervisor{Dr.-Ing. Holger Jäkel}
+\thesisSupervisor{Name of assistant}
 \thesisStartDate{24.10.2022}
 \thesisEndDate{24.04.2023}
 \thesisSignatureDate{Signature date} % TODO: Signature date
@@ -218,7 +218,7 @@
     \include{chapters/proximal_decoding}
     \include{chapters/lp_dec_using_admm}
     \include{chapters/comparison}
-%    \include{chapters/discussion}
+    \include{chapters/discussion}
     \include{chapters/conclusion}
     \include{chapters/appendix}